Geometry

12 pages, 281 KB

Open AccessFeature PaperArticle

Geometric Reinterpretation of Partial Differential Equations and Applications

by Satyanad Kichenassamy

Geometry 2026, 3(1), 6; https://doi.org/10.3390/geometry3010006 - 13 Mar 2026

We obtain improved regularity estimates on solutions of partial differential equations by combining the method of Fuchsian Reduction with geometric transformations. Examples include the meron problem and the regularity of the conformal radius. In each case, Reduction needs to be combined with a [...] Read more.

We obtain improved regularity estimates on solutions of partial differential equations by combining the method of Fuchsian Reduction with geometric transformations. Examples include the meron problem and the regularity of the conformal radius. In each case, Reduction needs to be combined with a reinterpretation of the underlying geometry. We argue that the geometric meaning assigned to a problem has an influence, positive or negative, on the range of methods envisioned for its solution, and that the Euler–Poisson–Darboux (EPD) equation cannot be properly understood within a single geometric framework. This explains the central position of EPD-like equations. Full article

10 pages, 299 KB

Open AccessArticle

Focal Circular Cubic and Complex Numbers

by Jiří Blažek

Geometry 2026, 3(1), 5; https://doi.org/10.3390/geometry3010005 - 10 Mar 2026

Abstract

A focal circular cubic is a locus of the foci of conics tangent to a given quadrilateral. In this article, we derive some of the focal curve’s basic properties. We study the curve in the complex plane and prove that the complex coordinates [...] Read more.

A focal circular cubic is a locus of the foci of conics tangent to a given quadrilateral. In this article, we derive some of the focal curve’s basic properties. We study the curve in the complex plane and prove that the complex coordinates of pairs of foci satisfy a quadratic equation. This equation can be expressed as a linear combination of two basic quadratic equations, which form a basis of a vector space. Furthermore, we give a nonstandard analytical condition, expressed in complex numbers, under which a circle can be inscribed in a quadrilateral. Finally, we leave the complex plane and show the construction of an arbitrary pair of foci of the curve by Euclidean means. Full article

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14 pages, 438 KB

Open AccessArticle

Properties of Elliptic Cycloids

by Matthew A. Pons and Nicholas D. White

Geometry 2026, 3(1), 4; https://doi.org/10.3390/geometry3010004 - 9 Feb 2026

Abstract

Given an ellipse rolling along a straight line without slipping and a point P on the ellipse, we will determine the shape of the elliptic cycloid traced by P as the ellipse rolls and compute the area under one arch of the elliptic [...] Read more.

Given an ellipse rolling along a straight line without slipping and a point P on the ellipse, we will determine the shape of the elliptic cycloid traced by P as the ellipse rolls and compute the area under one arch of the elliptic cycloid. We also investigate the arc length, though we are only able to express it as an integral. Full article

(This article belongs to the Special Issue Feature Papers in Geometry)

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13 pages, 577 KB

Open AccessFeature PaperArticle

Family of Non-Minkowski Measurable Fractals in

R^{2}

by Uta Freiberg and Jonas Lippold

Geometry 2026, 3(1), 3; https://doi.org/10.3390/geometry3010003 - 2 Feb 2026

Abstract

A long-standing conjecture of Lapidus asserts that under certain conditions a self-similar fractal set is not Minkowski measurable if and only if it is of lattice-type. For self-similar sets in

R

, the Lapidus conjecture has been confirmed. However, in higher dimensions, it [...] Read more.

A long-standing conjecture of Lapidus asserts that under certain conditions a self-similar fractal set is not Minkowski measurable if and only if it is of lattice-type. For self-similar sets in

R

, the Lapidus conjecture has been confirmed. However, in higher dimensions, it remains unclear whether all lattice-type self-similar sets are not Minkowski measurable. This work presents a family of lattice-type subsets in

R^{2}

that are not Minkowski measurable, hence providing further support for the conjecture. Furthermore, an argument is presented to illustrate why these sets are not covered by previous results. Full article

(This article belongs to the Special Issue Feature Papers in Geometry)

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15 pages, 302 KB

Open AccessArticle

Witten Deformation and Divergence-Free Symmetric Killing 2-Tensors

by Kwangho Choi and Junho Lee

Geometry 2026, 3(1), 2; https://doi.org/10.3390/geometry3010002 - 13 Jan 2026

Abstract

By using a Morse function and a Witten deformation argument, we obtain an upper bound for the dimensions of the space of divergence-free symmetric Killing p-tensors on a closed Riemannian manifold and explicitly calculate it for

p = 2

. Full article

(This article belongs to the Special Issue Feature Papers in Geometry)

13 pages, 1033 KB

Open AccessArticle

A New Characterization of the Symmedian Point

by Yu Chen and Robert J. Fisher

Geometry 2026, 3(1), 1; https://doi.org/10.3390/geometry3010001 - 8 Jan 2026

Abstract

Let

▵ A B C

be a triangle in the plane

E

, K be its symmedian point, and

C

be its circumcircle. Assume that P is a point on

C

such that it is not A, B, or C, [...] Read more.

Let

▵ A B C

be a triangle in the plane

E

, K be its symmedian point, and

C

be its circumcircle. Assume that P is a point on

C

such that it is not A, B, or C, it does not lie on the medians of

▵ A B C

, and the lines

ℓ_{A P}

,

ℓ_{B P}

, and

ℓ_{C P}

intersect

ℓ_{B C}

,

ℓ_{C A}

, and

ℓ_{A B}

at points

P_{a}

,

P_{b}

, and

P_{c}

, respectively. By the Ceva Concurrence Theorem, the harmonic conjugates

Q_{a}

,

Q_{b}

, and

Q_{c}

of

P_{a}

,

P_{b}

, and

P_{c}

in

\bar{B C}

,

\bar{C A}

, and

\bar{A B}

, respectively, are collinear. We prove that K lies on the line through

Q_{a}

,

Q_{b}

, and

Q_{c}

, which provides a new characterization of the symmedian point. Moreover, this one-to-one correspondence extends to a bijection from the entire circumcircle

C

onto the set

L_{K}

of all the lines in

E

that pass through K. Full article

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15 pages, 424 KB

Open AccessArticle

Some Nice Configurations of Golden Triangles

by Aldo Scimone

Geometry 2025, 2(4), 21; https://doi.org/10.3390/geometry2040021 - 10 Dec 2025

Abstract

It is well known among geometry scholars that the golden triangle, an isosceles triangle with sides and base in golden ratio, maintains a significant relationship with regular polygons, notably the regular pentagon, pentagram, and decagon. Extensive mathematical literature addresses this subject. Furthermore, its [...] Read more.

It is well known among geometry scholars that the golden triangle, an isosceles triangle with sides and base in golden ratio, maintains a significant relationship with regular polygons, notably the regular pentagon, pentagram, and decagon. Extensive mathematical literature addresses this subject. Furthermore, its close association with the golden ratio—a mathematical concept describing a harmonious and proportionate relationship between segments—renders it a noteworthy element in the fields of geometry, art, and architecture. Nevertheless, the interrelationships among these mathematical constructs frequently reveal unexpected configurations, thereby accentuating intriguing patterns. The purpose of this investigation is to highlight these novel configurations, which indicate new connections between the golden triangle and regular polygons. Full article

(This article belongs to the Special Issue Feature Papers in Geometry)

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8 pages, 217 KB

Open AccessArticle

A Simple Proof of Chen–Ricci Inequality and Applications

by Ion Mihai

Geometry 2025, 2(4), 20; https://doi.org/10.3390/geometry2040020 - 1 Dec 2025

Abstract

In the present paper, we give a simple proof of the Chen–Ricci inequality for submanifolds in Riemannian and Lorentzian space forms, respectively. Moreover, we extend the Chen–Ricci inequality to submanifolds in Lorentzian manifolds with a semi-symmetric non-metric connection. Full article

14 pages, 318 KB

Open AccessArticle

Side-Angle-Side, Side-Angle-Angle, and the Non-Continuous Absolute Plane

by John Donnelly

Geometry 2025, 2(4), 19; https://doi.org/10.3390/geometry2040019 - 17 Nov 2025

Abstract

We prove that if we start with a non-continuous absolute plane, remove Side-Angle-Side as an axiom, and replace it with the three new axioms, Side-Angle-Angle, angle addition, and the existence of angle bisectors, then the result is also an absolute plane. Full article

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22 pages, 1109 KB

Open AccessArticle

Drapeability and Λ-Frames

by Yevgenya Movshovich and John Wetzel

Geometry 2025, 2(4), 18; https://doi.org/10.3390/geometry2040018 - 4 Nov 2025

Abstract

In recent years, two quite different tools have been employed to study global properties of arcs in the plane. The first is drapeability, which grew from ideas of J. R. Alexander in early 2000s defining an arc drapeable if it lies in the [...] Read more.

In recent years, two quite different tools have been employed to study global properties of arcs in the plane. The first is drapeability, which grew from ideas of J. R. Alexander in early 2000s defining an arc drapeable if it lies in the convex hull of a shorter convex arc. The second is

Λ

-configuration, where an arc travels from one line to another and back. We investigate interrelations between these notions and in the process find drapeability criteria for open arcs, necessary and sufficient drapeability conditions for three-segment z-shaped arcs, and new bounds for the width of non-drapeable arcs. Full article

(This article belongs to the Special Issue Feature Papers in Geometry)

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32 pages, 355 KB

Open AccessArticle

Killing Vector Fields of Invariant Metrics

by Gerard Thompson

Geometry 2025, 2(4), 17; https://doi.org/10.3390/geometry2040017 - 22 Oct 2025

Cited by 1

Abstract

We study the existence of Killing vector fields for right-invariant metrics on low-dimensional Lie groups. Specifically, Lie groups of dimension two, three and four are considered. Before attempting to implement the differential conditions that comprise Killing’s equations, the metric is reduced as much [...] Read more.

We study the existence of Killing vector fields for right-invariant metrics on low-dimensional Lie groups. Specifically, Lie groups of dimension two, three and four are considered. Before attempting to implement the differential conditions that comprise Killing’s equations, the metric is reduced as much as possible by using the automorphism group of the Lie algebra. After revisiting the classification of the low-dimensional Lie algebras, we review some of the known results about Killing vector fields on Lie groups and add some new observations. Then we investigate indecomposable Lie algebras and attempt to solve Killing’s equations for each reduced metric. We introduce a matrix

M M

, that results from the integrability conditions of Killing’s equations. For

n = 4

, the matrix

M M

is of size

20 \times 6

. In the case where

M M

has maximal rank, for the Lie group problem considered in this article, only the left-invariant vector fields are Killing. The solution of Killing’s equations is performed by using MAPLE, and knowledge of the rank of

M M

can help to confirm that the solutions found by MAPLE are the only linearly independent solutions. After finding a maximal set of linearly independent solutions, the Lie algebra that they generate is identified to one in a standard list. Full article

38 pages, 5294 KB

Open AccessFeature PaperArticle

A Discrete Schwarzian Derivative via Circle Packing

by Kenneth Stephenson

Geometry 2025, 2(4), 16; https://doi.org/10.3390/geometry2040016 - 9 Oct 2025

Abstract

There exists an extensive and fairly comprehensive discrete analytic function theory which is based on circle packing. This paper introduces a faithful discrete analogue of the classical Schwarzian derivative to this theory and develops its basic properties. The motivation comes from the current [...] Read more.

There exists an extensive and fairly comprehensive discrete analytic function theory which is based on circle packing. This paper introduces a faithful discrete analogue of the classical Schwarzian derivative to this theory and develops its basic properties. The motivation comes from the current lack of circle packing algorithms in spherical geometry, and the discrete Schwarzian derivative may provide for new approaches. A companion localized notion called an intrinsic schwarzian is also investigated. The main concrete results of the paper are limited to circle packing flowers. A parameterization by intrinsic schwarzians is established, providing an essential packing criterion for flowers. The paper closes with the study of special classes of flowers that occur in the circle packing literature. As usual in circle packing, there are pleasant surprises at nearly every turn, so those not interested in circle packing theory may still enjoy the new and elementary geometry seen in these flowers. Full article

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11 pages, 326 KB

Open AccessArticle

The Sum of the Solid Angles of an n-Simplex

by Harold R. Parks and Dean C. Wills

Geometry 2025, 2(3), 15; https://doi.org/10.3390/geometry2030015 - 19 Sep 2025

Abstract

It was known in antiquity that the sum of the three angles of a triangle equals

π

. Surprisingly, it was not until 1952 that the corresponding question for a tetrahedron was addressed. In that year, J.W. Gaddum proved that the sum of [...] Read more.

It was known in antiquity that the sum of the three angles of a triangle equals

π

. Surprisingly, it was not until 1952 that the corresponding question for a tetrahedron was addressed. In that year, J.W. Gaddum proved that the sum of the four solid angles in a tetrahedron lies within the interval of

[0, 2 π]

and those lower and upper bounds are the best possible. In 2020, H. Katsuura showed that

2 π

was unachievable. In this paper, we generalize these results to show that for a non-degenerate n-simplex in

R^{n}

with

n \geq 3

, the solid angles at the vertices add up to a positive number that is less than one-half the

(n - 1)

-dimensional area of the unit sphere in

R^{n}

. We also show that there are examples for which the sum can be made arbitrarily close to the extreme values of 0 and one-half the

(n - 1)

-dimensional area of the unit sphere in

R^{n}

. Full article

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19 pages, 345 KB

Open AccessFeature PaperArticle

On d and M Problems for Newtonian Potentials in Euclidean n Space

by John Lewis

Geometry 2025, 2(3), 14; https://doi.org/10.3390/geometry2030014 - 2 Sep 2025

Abstract

In this paper, we first make and discuss a conjecture concerning Newtonian potentials in Euclidean n space which have all their mass on the unit sphere about the origin and are normalized to be one at the origin. The conjecture essentially divides these [...] Read more.

In this paper, we first make and discuss a conjecture concerning Newtonian potentials in Euclidean n space which have all their mass on the unit sphere about the origin and are normalized to be one at the origin. The conjecture essentially divides these potentials into subclasses whose criteria for membership is that a given member has its maximum on the closed unit ball at most M and its minimum at least d. It then lists the extremal potential in each subclass, which is conjectured to solve certain extremal problems. In Theorem 1, we show the existence of these extremal potentials. In Theorem 2, we prove an integral inequality on spheres about the origin, involving so-called extremal potentials, which lends credence to the conjecture. Full article

(This article belongs to the Special Issue Feature Papers in Geometry)

8 pages, 284 KB

Open AccessArticle

Generalization of Napoleon–Barlotti Theorem

by Jiří Blažek and Pavel Pech

Geometry 2025, 2(3), 13; https://doi.org/10.3390/geometry2030013 - 19 Aug 2025

Abstract

The Napoleon–Barlotti theorem belongs to the family of theorems related to the Petr–Douglas–Neumann theorem. The Napoleon–Barlotti theorem states: On the sides of an affine-regular n-gon construct regular n-gons. Then the centers of these regular n-gons form a regular n-gon. [...] Read more.

The Napoleon–Barlotti theorem belongs to the family of theorems related to the Petr–Douglas–Neumann theorem. The Napoleon–Barlotti theorem states: On the sides of an affine-regular n-gon construct regular n-gons. Then the centers of these regular n-gons form a regular n-gon. In the paper we give a generalization of this theorem. Full article

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17 pages, 1159 KB

Open AccessFeature PaperArticle

The Largest Circle Enclosing n Interior Lattice Points

by Jianqiang Zhao

Geometry 2025, 2(3), 12; https://doi.org/10.3390/geometry2030012 - 11 Aug 2025

Abstract

In this paper, we propose a class of elementary plane geometry problems closely related to the title of this paper. Here, a circle is the one-dimensional curve bounding a disk. For any non-negative integer, a circle is called n-enclosing if it contains [...] Read more.

In this paper, we propose a class of elementary plane geometry problems closely related to the title of this paper. Here, a circle is the one-dimensional curve bounding a disk. For any non-negative integer, a circle is called n-enclosing if it contains exactly n lattice points on the

x y

-plane in its interior. In this paper, we are mainly interested in when the largest n-enclosing circle exists and what the largest radius is. We study the small integer cases by hand and extend to all

n < 1100

with the aid of a computer. We find that frequently such a circle does not exist, e.g., when

n = 5, 6

. We then show a few general results on these circles including some regularities among their radii and an easy criterion to determine exactly when the largest n-enclosing circles exist. Further, from numerical evidence, we conjecture that the set of integers whose largest enclosing circles exist is infinite, and so is its complementary in the set of non-negative integers. Throughout this paper, we present more mysteries/problems/conjectures than answers/solutions/theorems. In particular, we list many conjectures and some unsolved problems including possible higher-dimensional generalizations at the end of the last two sections. Full article

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17 pages, 1278 KB

Open AccessReview

The Multiple Utility of Kelvin’s Inversion

by Eleftherios Protopapas

Geometry 2025, 2(3), 11; https://doi.org/10.3390/geometry2030011 - 9 Jul 2025

Abstract

Inversion with respect to a unit sphere is a powerful tool when dealing with many problems in Mathematics. This inversion preserves harmonicity in

R^{2},

but it does not in

R^{n},

for

n > 2 .

Lord Kelvin overcame this [...] Read more.

Inversion with respect to a unit sphere is a powerful tool when dealing with many problems in Mathematics. This inversion preserves harmonicity in

R^{2},

but it does not in

R^{n},

for

n > 2 .

Lord Kelvin overcame this problem by defining a new (at the time) inversion, the so-called Kelvin’s inversion (or transformation). This inversion has many good properties, making it extremely useful in each case where the geometry of the original problem raises issues. But by using Kelvin’s inversion, these issues are transformed into easier ones, due to a simpler geometry. In this review paper, we study Kelvin’s inversion, deploying its basic properties. Moreover, we present some applications, where its use enables scientists to solve difficult problems in scattering, electrostaticity, thermoelasticity, potential theory and bioengineering. Full article

(This article belongs to the Special Issue Feature Papers in Geometry)

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9 pages, 609 KB

Open AccessFeature PaperArticle

On Yiu’s Equilateral Triangles Associated with a Kiepert Hyperbola

by Cherng-tiao Perng

Geometry 2025, 2(3), 10; https://doi.org/10.3390/geometry2030010 - 1 Jul 2025

Abstract

In 2014, Paul Yiu constructed two equilateral triangles inscribed in a Kiepert hyperbola associated with a reference triangle. It was asserted that each of the equilateral triangles is triply perspective with the reference triangle, and in each case, the corresponding three perspectors are [...] Read more.

In 2014, Paul Yiu constructed two equilateral triangles inscribed in a Kiepert hyperbola associated with a reference triangle. It was asserted that each of the equilateral triangles is triply perspective with the reference triangle, and in each case, the corresponding three perspectors are collinear. In this note, we provide proof of his assertions. Furthermore, as an analogue of Lemoine’s problem, we formulated and answered the question of how to recover the reference triangle given a Kiepert hyperbola, one of the two Fermat points and one vertex of the reference triangle. Full article

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43 pages, 497 KB

Open AccessFeature PaperArticle

Problems in Invariant Differential Operators on Homogeneous Manifolds

by Jae-Hyun Yang

Geometry 2025, 2(2), 9; https://doi.org/10.3390/geometry2020009 - 9 Jun 2025

Abstract

In this paper, we consider six homogeneous manifolds

G L (n, R) / O (n, R)

,

S L (n, R) / S O (n, R)

, [...] Read more.

In this paper, we consider six homogeneous manifolds

G L (n, R) / O (n, R)

,

S L (n, R) / S O (n, R)

,

S p (2 n, R) / U (n),

(G L (n, R) ⋉ R^{(m, n)}) / O (n, R)

,

(S L (n, R) ⋉ R^{(m, n)}) / S O (n, R),

(S p (2 n, R) ⋉ H_{R}^{(n, m)}) / (U (n) \times S (m, R)) .

They are homogeneous manifolds which are important geometrically and number theoretically. These first three spaces are well-known symmetric spaces and the other three are not symmetric spaces. It is well known that the algebra of invariant differential operators on a symmetric space is commutative. The algebras of invariant differential operators on these three non-symmetric spaces are not commutative and have complicated generators. We discuss invariant differential operators on these non-symmetric spaces and provide natural but difficult problems about invariant theory. Full article

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